Under \(\tilde{\mathbb {P}}\) , the observation process \(Y\)  is a scaled Brownian motion
independent of \(X\) ; additionally the law of the signal process \(X\)  under
\(\tilde{\mathbb {P}}\)  is the same as its law under \(\mathbb {P}\)The proof at this statement is an immediate application of Girsanov's theorem and follows in a similar manner with the proof of Proposition 3.13 in [1]}..
Moreover, for every \(f\)  defined on the signal path space, we have
the Bayes' formula
\(\mathbb {E}\left(f((X_s)_{s\in [0,t]})~\vert ~ {\cal F }_t\right)=\frac{\tilde{\mathbb {E}}\left(f((X_s)_{s\in [0,t]})~Z_t(X,Y)~\vert ~ {\cal F }_t\right)}{\tilde{\mathbb {E}}\left(Z_t(X,Y)~\vert ~ {\cal F }_t\right)}.\) 
